 x Pseudo-octave Encyclopedia
A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

whose frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

is not 2:1 (2.3:1 or 1.9:1, for example), that of the octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency. When used as a basis for an equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

, the pseudo-octave may also be called the Interval of Equivalence (IoE), the Repeat Ratio, and the nonoctave.

## Stretched octave

The stretched octave, for example 2.01:1, sounds out of tune when played with true harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

overtones, but in tune when played with tones whose overtones are stretched equivalently.

In piano tuning
Piano tuning
Piano tuning is the act of making minute adjustments to the tensions of the strings of a piano to properly align the intervals between their tones so that the instrument is in tune. The meaning of the term in tune in the context of piano tuning is not simply a particular fixed set of pitches...

, stretched octaves are commonly encountered, where the inharmonicity
Inharmonicity
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency....

caused by string thickness and tension makes it necessary to widen every interval very slightly.

The octaves of Bali
Bali
Bali is an Indonesian island located in the westernmost end of the Lesser Sunda Islands, lying between Java to the west and Lombok to the east...

nese gamelan
Gamelan
A gamelan is a musical ensemble from Indonesia, typically from the islands of Bali or Java, featuring a variety of instruments such as metallophones, xylophones, drums and gongs; bamboo flutes, bowed and plucked strings. Vocalists may also be included....

s are never tuned 2:1, but instead are stretched or compressed in a consistent manner throughout the range of each individual gamelan, due to the physical characteristics of their instruments. Another example is the tritave of the Bohlen–Pierce scale.

Stretched octaves are caused by the physics of standing wave
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...

s in a stretched wire. The pitch of each overtone
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...

produced by a piano string is determined by the ratio of the string's restoring force
Restoring force
Restoring force, in a physics context, is a variable force that gives rise to an equilibrium in a physical system. If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium....

(expressed as a spring constant), divided by its mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

per unit length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

.

In an ideal piano string, the only restoring force would be due to the tension in the string. In practice, piano strings are made from high-carbon steel
Steel
Steel is an alloy that consists mostly of iron and has a carbon content between 0.2% and 2.1% by weight, depending on the grade. Carbon is the most common alloying material for iron, but various other alloying elements are used, such as manganese, chromium, vanadium, and tungsten...

, which is stiff. The stiffness adds an extra restoring force to each string; the amount of this extra force depends on the amount of bend being induced in the string. Higher normal mode
Normal mode
A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...

s bend the string more, inducing more stiffness-related force and sharpening the pitch of the resulting overtone.

Octave stretching is less apparent on large pianos which have longer strings and hence less curvature for a given displacement
Displacement (vector)
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...

; that is one reason why orchestras go to the expense of using very long concert grand pianos rather than shorter, less expensive baby grand, upright, or spinet piano
Spinet
A spinet is a smaller type of harpsichord or other keyboard instrument, such as a piano or organ.-Spinets as harpsichords:While the term spinet is used to designate a harpsichord, typically what is meant is the bentside spinet, described in this section...

s. (The other reason is that long strings under high tension can store more acoustic energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

than can short strings, giving larger instruments more volume and better sustain than similar, smaller instruments). 